Problem: Simplify; express your answer in exponential form. Assume $a\neq 0, z\neq 0$. $\dfrac{{(a^{3})^{-3}}}{{(a^{3}z^{-3})^{-5}}}$
Answer: To start, try working on the numerator and the denominator independently. In the numerator, we have ${a^{3}}$ to the exponent ${-3}$ . Now ${3 \times -3 = -9}$ , so ${(a^{3})^{-3} = a^{-9}}$ In the denominator, we can use the distributive property of exponents. ${(a^{3}z^{-3})^{-5} = (a^{3})^{-5}(z^{-3})^{-5}}$ Simplify using the same method from the numerator and put the entire equation together. $\dfrac{{(a^{3})^{-3}}}{{(a^{3}z^{-3})^{-5}}} = \dfrac{{a^{-9}}}{{a^{-15}z^{15}}}$ Break up the equation by variable and simplify. $\dfrac{{a^{-9}}}{{a^{-15}z^{15}}} = \dfrac{{a^{-9}}}{{a^{-15}}} \cdot \dfrac{{1}}{{z^{15}}} = a^{{-9} - {(-15)}} \cdot z^{- {15}} = a^{6}z^{-15}$.